6520-2024-08

Carefully state Van Kampen’s theorem. The version for two subsets suffices. Use Van Kampen’s theorem to compute \(\pi _1(T^2 \vee T^2)\) where \(T^2\) is a torus.

Suppose that a complex \(X\) is the union of two subcomplexes \(A\) and \(B\) with \(A\), \(B\), and \(A \cap B\) contractible. Prove that \(X\) is contractible.

1. Show that \(\mathbb {R}P^2\) has a 2-fold cover which is \(S^2\).
2. Show that the Klein bottle has a 2-fold cover which is a torus.

We can regard \(\pi _1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0) \to \) \((X,x_0)\). Let \([S^1,X]\) be the set of homotopy classes of maps \(S^1 \to X\) with no condition on basepoints. Thus there is a natural map \(\psi : \pi _1(X,x_0) \to [S^1,X]\) obtained by ignoring basepoints. Show that \(\psi \) is onto if \(X\) is path connected.

Prove or give a (justified) counterexample. If the first homology group \(H_1(X)\) of a connected CW-complex \(X\) is trivial then \(X\) is contractible.

Compute the relative homology groups \(H_n(X,A)\) where \(X\) is \(S^2\) and \(A\) is a finite set of points in \(X\).

Show that if \(f : S^n \to S^n\) has degree \(d\) then the induced map \(f^* : H^n(S^n;G) \to H^n(S^n;G)\) is multiplication by \(d\).